Interpolating a DFT

Starting with a sampled spectrum $X(\omega_k)$ , $k=0,1,\ldots,N-1$ , typically obtained from a DFT, we can interpolate by taking the DTFT of the IDFT which is not periodically extended, but instead zero-padded [264]:^3.8

$$\begin{eqnarray*} X(\omega) &=& \hbox{\sc DTFT}(\hbox{\sc ZeroPad}_{\infty}(\hbox{\sc IDFT}_N(X)))\\ &\isdef & \sum_{n=-N/2}^{N/2-1}\left[\frac{1}{N}\sum_{k=0}^{N-1}X(\omega_k) e^{j\omega_k n}\right]e^{-j\omega n}\\ &=& \sum_{k=0}^{N-1}X(\omega_k) \left[\frac{1}{N}\sum_{n=-N/2}^{N/2-1} e^{j(\omega_k-\omega) n}\right]\\ &=& \sum_{k=0}^{N-1}X(\omega_k)\,\hbox{asinc}_N(\omega-\omega_k)\\ &=& \left<X,\hbox{\sc Sample}_N\{\hbox{\sc Shift}_{\omega}(\hbox{asinc}_N)\}\right> \end{eqnarray*}$$

(The aliased sinc function, $\hbox{asinc}_N(\omega)$ , is derived in §3.1.) Thus, zero-padding in the time domain interpolates a spectrum consisting of $N$ samples around the unit circle by means of `` $\hbox{asinc}_N$ interpolation.'' This is ideal, time-limited interpolation in the frequency domain using the aliased sinc function as an interpolation kernel. We can almost rewrite the last line above as $X(\omega)=(X\ast \hbox{asinc}_N)_\omega$ , but such an expression would normally be defined only for $\omega=\omega_l=2\pi l/N$ , where $l$ is some integer, since $X$ is discrete while $\hbox{asinc}_N$ is continuous.

Figure F.1 lists a matlab function for performing ideal spectral interpolation directly in the frequency domain. Such an approach is normally only used when non-uniform sampling of the frequency axis is needed. For uniform spectral upsampling, it is more typical to take an inverse FFT, zero pad, then a longer FFT, as discussed further in the next section.